def test1():
dim = 3
alpha = 1 / 2
beta = -1.
scale = 100.
A = make_positive_definite_matrix(scale=scale, dim=dim)
evals, U = get_eigens_for_positive_definite_matrix(A)
L = np.diag(evals)
assert np.allclose(A, U.dot(L).dot(U.T)), "Eigendecomposition is wrong"
b = scale * (np.random.rand(dim) * 2 - 1)
c = np.random.uniform(low=-scale, high=scale)
x0 = scale * (np.random.rand(dim) * 2 - 1)
SD_steps = steepest_descent(A, b, x0)
print("test1: number of steepest descent steps = {}".format(
SD_steps.shape[1]))
x0c = SD_steps[:, 0]
assert np.allclose(x0, x0c), "x0={} not equal to x0c={}".format(x0, x0c)
xs = SD_steps[:, -1]
_, xsc = get_scipy_mins_for_quadratic_form(x0, A, b, c, alpha, beta)
assert np.allclose(xs, xsc), "xs={} not equal to xsc={}".format(xs, xsc)
x1 = SD_steps[:, 1]
x2 = SD_steps[:, 2]
u = x1 - x0
v = xs - x1
w = x2 - x1
x = xs - x2
assert_A_orthogonal(A, u, v, atol=1e-5)
assert_A_orthogonal(A, w, x, atol=1e-5)
D_k, phi_k = compute_Dk_for_x(u, A, b, c, alpha, beta, evals, U)
D_q, phi_q = compute_Dk_for_x(v, A, b, c, alpha, beta, evals, U)
D_q_inv = np.linalg.inv(D_q)
W = U.dot(D_q_inv).dot(U.T)
print("D_q_inv={}".format(D_q_inv))
print("W={}".format(W))
a = W.dot(u)
b = W.dot(v)
assert_orthogonal(a, b)
print("a={} and b={} are orthogonal".format(a, b))
SD_steps_2 = steepest_descent(np.eye(dim), b, x)
print("test1: number of steepest descent steps = {}".format(
SD_steps_2.shape[1]))
ax0 = SD_steps_2[:, 0]
ax1 = SD_steps_2[:, -1]
print("x2={} xs={} ax0={} ax1={}".format(x2, xs, ax0, ax1))
w0 = A.dot(ax0)
w1 = A.dot(ax1)
w1 = A.dot(ax1 + b)
print("w0={} w1={}".format(w0, w1))
return
def figure_GSC_1(debug_print=False):
A = np_array([4., 1, 1, 15]).reshape(2, 2)
data1, data2, data3 = [], [], []
u = np_array([1., 1])
v = np_array([-0.5, 2])
w0, w1 = gram_schmidt_conjugation(A, [u, v])
assert_A_orthogonal(A, w0, w1)
c = 2.3
u0 = w0
u1 = w1 + c * w0
du0, m1 = draw_vector_R2(u0, color='blue', name=py_text_sub('u', 0))
du1, m2 = draw_vector_R2(u1, color='blue', name=py_text_sub('u', 1))
data1.extend(du0)
data1.extend(du1)
dw0, m1 = draw_vector_R2(u1, t=w1, color='red')
dw1, m2 = draw_vector_R2(w1, color='green')
Au0 = A.dot(u0)
Au0n = Au0 / norm(Au0)
dAu0, m1 = draw_vector_R2(3.5 * Au0n,
color='grey',
name=py_text_sub('Au', 0))
data2.extend(dw0)
data2.extend(dw1)
data2.extend(du0)
data2.extend(du1)
data2.extend(dAu0)
d0, d1 = gram_schmidt_conjugation(A, [u0, u1])
assert_A_orthogonal(A, d0, d1)
dd0, m1 = draw_vector_R2(d0, color='blue', name=py_text_sub('d', '(0)'))
dd1, m2 = draw_vector_R2(d1, color='blue', name=py_text_sub('d', '(1)'))
data3.extend(dd0)
data3.extend(dd1)
m = np_max([m1, m2])
plot_it_R2_short_3in1(data1,
data2,
data3,
axes_max1=m,
axes_max2=m,
axes_max3=m,
title='GSC.Fig.1',
filename='Fig_GSC_1.html')
return
def figure_ISD_1(debug_print=False):
steps = steepest_descent(A, b, x0)
assert np_allclose(steps[:, -1], center.flatten()
), "steepest_descent didn't find the minimum point of f"
for i in range(steps.shape[1] - 1):
x_i = steps[:, i]
r_i = b.flatten() - A.dot(x_i)
x_ip1 = steps[:, i + 1]
e_ip1 = x_ip1 - center.flatten()
assert_A_orthogonal(A, r_i, e_ip1)
text = [py_text_sub('x', '(0)'), py_text_sub('x', '(1)')]
t2 = [''] * (steps.shape[1] - 3)
text.extend(t2)
text.append(py_text_sub('x', '(*)'))
textposition = ['middle left', 'top right']
tp2 = ['top center'] * len(t2)
textposition.extend(tp2)
textposition.append('bottom center')
plotly_data = []
r0 = b - A.dot(x0)
alpha0 = innprd(r0, r0) / innprd(r0, A.dot(r0))
x1 = x0 + alpha0 * r0
r0_normalized = r0.flatten() / norm(r0)
d, _ = qfc.prep_vector_data(r0_normalized,
center=x0,
M=None,
color='rgb(0,140,0)',
name=py_text_sub('r', '(0)'))
plotly_data.extend(d)
e1 = x1 - center
d, _ = qfc.prep_vector_data(e1,
center=center,
M=None,
color='rgb(0,140,0)',
name=py_text_sub('e', '(1)'),
textposition='bottom left')
plotly_data.extend(d)
draw_concentric_contours(steps,
text,
textposition,
plotly_data=plotly_data,
title='ISD.Fig.1',
filename='Fig_ISD_1.html',
debug_print=debug_print)
return
def figure_CJDR_5(debug_print=False):
steps = compute_steps_in_ball_space()
for i in [0, 1]:
steps[:, i] = (M.dot(steps[:, i].reshape(2, 1) - center) +
center).reshape(2, )
u = steps[:, 1] - steps[:, 0]
v = steps[:, 2] - steps[:, 1]
assert_A_orthogonal(A, u, v)
text = [
py_text_sub('x', '(0)'),
py_text_sub('x', '(1)'),
py_text_sub('x', '(*)')
]
textposition = ['middle left', 'top center', 'bottom center']
draw_concentric_contours(steps,
text,
textposition,
title='CJDR.Fig.5',
filename='Fig_CJDR_5.html',
debug_print=debug_print)
return
def figure_CJDR_1(debug_print=False):
r0 = b - A.dot(x0)
alpha0 = innprd(r0, r0) / innprd(r0, A.dot(r0))
x1 = x0 + alpha0 * r0
steps = np_hstack((x0, x1, center))
text = [
py_text_sub('x', '(0)'),
py_text_sub('x', '(1)'),
py_text_sub('x', '(*)')
]
textposition = ['middle left', 'middle left', 'bottom center']
plotly_data = []
r0_normalized = r0.flatten() / norm(r0)
d, _ = qfc.prep_vector_data(r0_normalized,
center=x0,
M=None,
color='rgb(0,140,0)',
name=py_text_sub('r', '(0)'))
plotly_data.extend(d)
Ar0 = A.dot(r0_normalized)
d, _ = qfc.prep_vector_data(Ar0,
center=x0,
M=None,
color='rgb(170,0,0)',
name=py_text_sub('Ar', '(0)'))
plotly_data.extend(d)
e1 = x1 - center
d, _ = qfc.prep_vector_data(e1,
center=center,
M=None,
color='rgb(0,140,0)',
name=py_text_sub('e', '(1)'),
textposition='top right')
plotly_data.extend(d)
e1_normalized = e1.flatten() / norm(e1)
d, _ = qfc.prep_vector_data(e1_normalized,
center=x0,
M=None,
color='rgb(170,0,0)',
name=py_text_sub('e', '(1)'))
plotly_data.extend(d)
assert_A_orthogonal(A, r0, -e1)
d, _ = qfc.prep_vector_data(-e1_normalized,
center=x0,
M=None,
color='rgb(170,0,0)',
name=py_text_sub('-e', '(1)'))
plotly_data.extend(d)
draw_concentric_contours(steps,
text,
textposition,
plotly_data=plotly_data,
title='CJDR.Fig.1',
filename='Fig_CJDR_1.html',
steps_mode='text, markers',
debug_print=debug_print)
return